Question

Sketch the graph of the inequality. \(x+y<9\)

Short answer

The graph of this inequality is a line with slope -1 and y-intercept 9, excluding the line itself, and shading the area beneath it.

Step-by-step solution

Treating as Equation

Treat the inequality as an equation, i.e., \(x + y = 9\). Then plot two points that satisfy this equation, for instance when \(x=0\) then \(y=9\) and when \(y=0\) then \(x=9\). This will help in drawing a solid line representing the equality.

Deciding Which Area to Shade

The inequality symbol '<' suggests that the solution is not on the line but in the area beneath or above it. To decide which area, select a test point not on the line, e.g., origin (0,0) which makes \(x + y = 0\). As 0 is less than 9, the inequality is satisfied, thus shading the area that includes the origin.

Shading the Area

Shade the half-plane that is beneath the line and includes the origin. Any point in this shaded area will satisfy the inequality \(x + y < 9\).

Checking Your Solution

To confirm the solution, select another point in the shaded area and verify that it satisfies the inequality.